Question

A patient is classified as having gestational diabetes if their average glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca's doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ= 143 mg/dl and σ = 4 mg/dl. Using the Central Limit Theorem, determine the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured:

1. Once?
2. 5
3. 7

Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.

Answer 1

We have:

x = 140 milligrams per deciliter (mg/dl)

μ= 143 mg/dl

σ = 4 mg/dl

Once?

The test statistic, z = (x - µ)/σ

z = (140 - 143)/4

z = -0.75

P(z > -0.75) = 0.7734

The probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured is 0.7734.

5

n = 5

The test statistic, z = (x - µ)/σ/√n

z = (140 - 143)/4/√5

z = -1.68

P(z > -1.68) = 0.9532

The probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured is 0.9532.

7

n = 7

The test statistic, z = (x - µ)/σ/√n

z = (140 - 143)/4/√7

z = -1.98

P(z > -1.98) = 0.9764

The probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured is 0.9764.

Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.

As the sample size increases, the standard error decreases, so, the probability of Rebecca being diagnosed with gestational diabetes increases.